Injured Hypercube Research Articles

Fault-tolerant routing methodology for hypercube and cube-connected cycles interconnection networks

This paper presents a fault-tolerant routing methodology for both injured hypercube and cube-connected cycles interconnection topologies. The proposed routing methodology efficiently tolerates any pattern of faulty regions with any number of faulty nodes in the network which is based on the best-first search and backtracking strategy. Deadlock freedom of the proposed routing methodology is obtained by only one virtual channel per physical channel. In order to evaluate the proposed routing methodology, a 7-dimensional hypercube network is simulated in various conditions, i.e., different traffic rates, different number of faulty nodes and different message lengths. Simulation results confirm that the proposed routing methodology in comparison with the previous methods provides acceptable performance while it significantly increases the reliability of the network. It also guarantees delivery of messages between any pair of source and destination while the network is connected.

On the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty hypercube

Hsieh and Yu (2007) first claimed that an injured n-dimensional hypercube Q n contains (n?1?f)-mutually independent fault-free Hamiltonian cycles, where f≤n?2 denotes the total number of permanent edge-faults in Q n for n?4, and edge-faults can occur everywhere at random. Later, Kueng et al. (2009a) presented a formal proof to validate Hsieh and Yu's argument. This paper aims to improve this mentioned result by showing that up to (n?f)-mutually independent fault-free Hamiltonian cycles can be embedded under the same condition. Let F denote the set of f faulty edges. If all faulty edges happen to be incident with an identical vertex s, i.e., the minimum degree of the survival graph Q n ?F is equal to n?f, then Q n ?F contains at most (n?f)-mutually independent Hamiltonian cycles starting from s. From such a point of view, the presented result is optimal. Thus, not only does our improvement increase the number of mutually independent fault-free Hamiltonian cycles by one, but also the optimality can be achieved.

Highly fault-tolerant cycle embeddings of hypercubes

The hypercube Q n is one of the most popular networks. In this paper, we first prove that the n -dimensional hypercube is 2 n − 5 conditional fault-bipancyclic. That is, an injured hypercube with up to 2 n − 5 faulty links has a cycle of length l for every even 4 ⩽ l ⩽ 2 n when each node of the hypercube is incident with at least two healthy links. In addition, if a certain node is incident with less than two healthy links, we show that an injured hypercube contains cycles of all even lengths except hamiltonian cycles with up to 2 n − 3 faulty links. Furthermore, the above two results are optimal. In conclusion, we find cycles of all possible lengths in injured hypercubes with up to 2 n − 5 faulty links under all possible fault distributions.

Optimal broadcasting in injured hypercubes using directed safety levels

Reliable communication in injured hypercubes with faulty links/nodes using directed safety levels is studied in this paper. In this approach, each node u in an n-dimensional hypercube ( n-cube) is associated with a sequence of directed safety levels. A directed safety level associated with node u is an approximated measure of broadcast capability of performing optimal broadcasting from u in an ( n−1)-subcube containing u. By optimal broadcasting, we mean that the broadcast message reaches each destination through a shortest path (i.e., the length of each path is equal to the Hamming distance between the source and destination). Directed safety levels are based on a special coding scheme generalized from Wu's safety level model and are calculated through n−1 rounds of information exchanges and updates among neighboring nodes. In this model, fault information of nodes within distance- d is precisely represented to better capture link faults, whereas fault information of nodes outside distance- d is approximated as in the regular safety level model. Optimal broadcasting at node u is guaranteed if node u is globally safe, which is defined in terms of the directed safety levels associated with u. The directed safety level model with d=1 or d=2 has the same asymptotic complexity as the regular safety level model. Simulation results show a significant improvement in terms of optimal broadcast capability in injured hypercubes.

Fault-tolerant routing in hypercube multicomputers using local safety information

This paper studies fault-tolerant routing for injured hypercubes using local safety information. It is shown that a minimum feasible path is always available if the spanning subcube that contains both source and destination is safe. The safety information outside the spanning subcube is applied only when derouting is needed. A routing scheme based on local safety information is proposed and the extra cost to obtain local safety information is comparable to the one based on global safety information. The proposed algorithm guarantees to find a minimum feasible path if the spanning subcube is contained in a maximal safe subcube and the source is locally safe in the maximal safe subcube. A new technique to set up a partial path is proposed based on local safety information when the above conditions are not met. Sufficient simulation results are provided to demonstrate the effectiveness of the method by comparing with the previous methods.

Tolerating faults in injured hypercubes using maximal fault-free subcube-ring

In this paper, we present a reconfiguration approach to identify the maximal fault-free subcube-ring for tolerating faults in injured hypercubes. The fault-free subcube-ring is connected by a ring of fault-free subcubes with dilation 3. By exploiting the size of fault-free subcubes as large as possible, the maximal fault-free subcube-ring with higher processor utilization is obtained. Using this approach, we can tolerate more than n faults in n-dimensional hypercubes. To demonstrate the fault-tolerant capability of our approach, we implement two fault-tolerant algorithms, matrix-multiplication and sorting algorithms, on the nCUBE/2E hypercube machine with 32 processors. The simulation results show that our reconfiguration approach has low performance slowdown and high processor utilization.

Optimal broadcasting in hypercubes with link faults using limited global information

Abstract We propose a fault-tolerant broadcasting algorithm for hypercubes with link faults. This algorithm is based on an extended spanning binomial tree structure that still keeps the simplicity of conventional binomial-tree-based broadcasting. In addition, it is optimal in the sense that exactly n steps are required to complete a broadcast in an n-dimensional injured hypercube with up to n − 2 faulty links. We also show that n − 2 is the maximum number of faulty links that can be tolerated in any optimal broadcast scheme in an n-dimensional hypercube. To implement the proposed algorithm each node keeps information of nearby faulty links in terms of addresses for those faulty adjacent m-subcubes that contain at least m − 1 faulty links.

On the Embedding of a Class of Regular Graphs in a Faulty Hypercube

In a hypercube of high dimension, the interconnection is quite complex and thus links are likely to fail. In this paper, a general, systematic solution is proposed for the embedding of a group of regular graphs in an injured hypercube with faulty links. These graphs include rings, linear paths, binomial trees, binary trees, meshes, tori, products of the above graphs, and many others. Unlike many existing algorithms which are capable of embedding only one type of graphs, our algorithm embeds the above graphs in a unified way, all centered around a notion callededge matrix.

Safety levels-an efficient mechanism for achieving reliable broadcasting in hypercubes

We consider a distributed broadcasting algorithm for injured hypercubes using incomplete spanning binomial trees. An injured hypercube is a connected hypercube with faulty nodes. The incomplete spanning binomial tree proposed in this paper is a useful structure for implementing broadcasting in injured hypercubes. It is defined as a sub-tree of a regular spanning binomial tree that connects all the nonfaulty nodes. We show that in an injured n-dimensional hypercube with m faulty nodes, there are at least 2/sup n/-2/sup m/ source nodes (called l-nodes), each of which can generate an incomplete spanning binomial tree. A method is proposed to locate a large subset of the l-node set using the concept of safety level. The safety level of each node in an n-dimensional hypercube can be easily calculated through n-1 rounds of information exchange among neighboring nodes. An optimal broadcast initiated from a safe node is proposed. When a nonfaulty source node is unsafe and there are at most n-1 faulty nodes in an injured n-dimensional hypercube, the proposed broadcasting scheme requires at most n+1 steps. >

An Optimal Fault-Tolerant Nonredundant Broadcasting Scheme in Injured Hypercubes

We propose a nonredundant broadcasting scheme for injured hypercube multicomputers with direct-connection capability, that is, where each hypercube node has a separate router. The proposed scheme is first presented to cover faulty links only. Under the assumption that no more than n − 1 faulty links exist in an injured n-cube, the proposed scheme is optimal in the sense that it generates a spanning binomial tree from a given node whenever there exists such a tree originated from that node. The scheme is also extended to cover both node and link faults. Although the extended scheme is no longer optimal, we show that by using the extended scheme the maximum number of routing steps required from the source node s to any other node a is H( s, a) + 2, where H stands for the Hamming distance. The proposed scheme uses global network information. Evaluation studies show that the proposed scheme achieves optimal broadcasting a high percentage of times.

Deadlock-free fault-tolerant routing in injured hypercubes

Wormhole routing with the e-cube algorithm is an excellent solution for deadlock-free interprocess communication in healthy hypercubes. However, it does not work for injured hypercubes where some nodes and/or links are faulty. The authors propose a new deadlock-free routing scheme in an injured hypercube with the wormhole routing capability. All previously proposed schemes suggest the use of virtual channels to avoid the cycle of resource dependency. By contrast, the authors' scheme is based on the re-establishment of a routing path to the destination, but it does not always yield a shortest path between the source and destination. The proposed routing scheme uses either wormhole routing or staged routing, depending on the availability of one or more healthy (n-2)-cubes within an injured n-cube. >

Adaptive fault-tolerant routing in hypercube multicomputers

A connected hypercube with faulty links and/or nodes is called an injured hypercube. A distributed adaptive fault-tolerant routing scheme is proposed for an injured hypercube in which each node is required to know only the condition of its own links. Despite its simplicity, this scheme is shown to be capable of routing messages successfully in an injured n-dimensional hypercube as long as the number of faulty components is less than n. Moreover, it is proved that this scheme routes messages via shortest paths with a rather high probability, and the expected length of a resulting path is very close so that of a shortest path. Since the assumption that the number of faulty components is less than n in an n-dimensional hypercube might limit the usefulness of the above scheme, a routing scheme based on depth-first search which works in the presence of an arbitrary number of faulty components is introduced. Due to the insufficient information on faulty components, however, the paths chosen by this scheme may not always be the shortest. To guarantee all messages to be routed via shortest paths, the authors propose to equip every node with more information than that on its own links. The effects of this additional information on routing efficiency are analyzed, and the additional information to be kept at each node for the shortest path routing is determined. Several examples and remarks are given to illustrate the results. >